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Cambridge IGCSE0607

Fractions, decimals and percentages

International Mathematics 0607 Chapter Notes

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Fractions, decimals and percentages
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Fractions, decimals and percentages notes

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In this chapter

  1. 1Proper, improper and mixed fractions
  2. 2Adding, subtracting, multiplying and dividing
  3. 3Fractions of an amount and finding the whole
  4. 4Converting between fractions, decimals and percentages
  5. 5Equivalent forms and comparing

1. Proper, improper and mixed fractions

A proper fraction is less than 1, such as 3/4. An improper fraction is 1 or more, such as 31/7. A mixed number is a whole number and a fraction, such as 43/7.

To write an improper fraction as a mixed number, divide the numerator by the denominator. The quotient is the whole number and the remainder is the new numerator.

To write a fraction in its simplest form, divide the top and bottom by their highest common factor.

31/7 = 43/7

ab/c = (a × c + b)/c

Examiner insight

“Most answers were correct, but a few candidates were unable to give their answer as a mixed number.”

Worked example 13 marks

Work out 15/6 + 31/4. Give your answer as a mixed number in its simplest form.

  1. 1

    improper fractions: 11/6 + 13/4 [1]

  2. 2

    common denominator 12: 22/12 + 39/12 = 61/12 [1]

  3. 3

    = 51/12 [1]

2. Adding, subtracting, multiplying and dividing

To add or subtract, change mixed numbers to improper fractions and use a common denominator. Add or subtract the numerators only. The denominator does not change.

To multiply, multiply the numerators and multiply the denominators. Cancel first if you can.

To divide, turn the second fraction upside down (use its reciprocal) and multiply. A whole number such as 5 is 5/1.

a/b ± c/d = (ad ± bc)/bd

a/b × c/d = ac/bd

a/b ÷ c/d = a/b × d/c

Examiner insight

“Some converted the mixed fractions correctly and then used an incorrect method to do the subtraction” “Some candidates made numerical slips in an otherwise correct method.”

Worked example 13 marks

Work out 21/5 - 12/3.

  1. 1

    improper fractions: 11/5 - 5/3 [1]

  2. 2

    common denominator 15: 33/15 - 25/15 [1]

  3. 3

    = 8/15 [1]

3. Fractions of an amount and finding the whole

To find a fraction of an amount, divide by the denominator, then multiply by the numerator. So 5/6 of 42 is 42 ÷ 6 × 5 = 35.

To find the whole from a part, divide by the numerator to find one share, then multiply by the denominator.

When the answer is a time, change it to hours and minutes at the end. There are 60 minutes in 1 hour.

whole = part ÷ numerator × denominator

Examiner insight

“A few candidates converted the correct answer incorrectly into hours and minutes.”

Worked example 13 marks

Kaya watches 25 minutes of a film. This is 5/26 of the film. Work out how long the whole film lasts. Give your answer in hours and minutes.

  1. 1

    5/26 is 25 minutes, so 1/26 is 25 ÷ 5 = 5 minutes [1]

  2. 2

    whole film = 5 × 26 = 130 minutes [1]

  3. 3

    130 = 2 × 60 + 10, so 2 h 10 min [1]

4. Converting between fractions, decimals and percentages

Percent means out of 100. So 30% = 30/100, which is 3/10 in simplest form, and 0.15 = 15%.

To write a fraction as a decimal, divide the numerator by the denominator. To write a fraction as a percentage, multiply the decimal by 100.

To write a decimal as a fraction, put it over 10, 100 or 1000. So 0.17 = 17/100.

To write a percentage as a decimal, divide by 100. So 65% = 0.65, which is 13/20 as a fraction.

3/5 = 3 ÷ 5 = 0.6

0.6 × 100 = 60%

Examiner insight

“There were many correct answers but some candidates did not read the question clearly enough and wrote the fraction as a decimal correct to 3 decimal places, rather than a percentage correct to 3 decimal places.”

Worked example 12 marks

Write 3/5(a) as a decimal(b) as a percentage.

  1. 1

    (a) 3 ÷ 5 = 0.6 [1]

  2. 2

    (b) 0.6 × 100 = 60% [1]

5. Equivalent forms and comparing

Equivalent fractions have the same value, such as 3/8 and 6/16. Multiply or divide the top and bottom by the same number.

To compare amounts given in different forms, write them all in the same form. Then compare, or work each out as a number.

A fraction needs a whole-number numerator and denominator. A decimal in the denominator does not earn the mark.

Examiner insight

“Some gave fractions where the denominator was a decimal which did not get a mark.”

Worked example 13 marks

A: 16% of 360 dollars. B: 2/9 of 360 dollars. Work out which is greater, A or B, and by how much. Show how you decide.

  1. 1

    A = 0.16 × 360 = 57.60 dollars [1]

  2. 2

    B = 360 ÷ 9 × 2 = 80 dollars [1]

  3. 3

    B is greater by 80 - 57.60 = 22.40 dollars [1]

Recap

  • Proper fraction: less than 1. Improper fraction: 1 or more. Mixed number: whole number plus fraction.
  • Simplest form: divide top and bottom by their highest common factor.
  • Add and subtract with a common denominator. Divide by multiplying by the reciprocal.
  • Whole = part ÷ numerator × denominator.
  • Fraction to decimal: top ÷ bottom. Decimal to percentage: × 100.
  • Compare amounts in the same form, and answer in the form the question asks for.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Write 17/5 as a mixed number.1 mark
  2. Write 36/48 in its simplest form.1 mark
  3. Do not use a calculator. Write 0.35 as a fraction in its simplest form.2 marks
  4. Write 3/8 as a percentage.2 marks
  5. Do not use a calculator. Work out 2/3 + 3/4. Give your answer as a mixed number.3 marks
  6. Do not use a calculator. Work out 31/2 ÷ 13/4.3 marks
  7. Do not use a calculator. Put these in order, smallest first: 0.6, 5/8, 59%.3 marks
  8. A jacket is on sale for 48 dollars. This is 3/5 of its original price. Find the original price.2 marks
  9. Do not use a calculator. A phone battery is fully charged. Streaming a film uses 3/8 of the full charge. A game uses 0.35 of the full charge. The charge that is left runs phone calls for 77 minutes. Calls use charge at a steady rate. Work out how long a full charge runs phone calls. Give your answer in hours and minutes.5 marks
  10. A school has a budget for a trip. Transport takes 2/5 of the budget. Food takes 35% of the money left after transport is paid. The rest of the budget, 468 dollars, pays for tickets. Find the budget. Then write the cost of food as a fraction of the cost of transport, in its simplest form.6 marks

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